2015
\[\begin{align*} \\ &\text{HPH gene $g$ if} \ &&\mu_{g, \ \text{hybrid}} > \max \left ( \mu_{g, \ \text{parent 1}}, \ \mu_{g, \ \text{parent 2}}\right ) \\ &\text{LPH if} &&\mu_{g, \ \text{hybrid}} < \min \left ( \mu_{g, \ \text{parent 1}}, \ \mu_{g, \ \text{parent 2}}\right ) \\ \\ \text{Equ}&\text{ivalently,} \\ \\ &\text{HPH if} &&\alpha_g + \delta_g > \ \ \ |\alpha_g - \delta_g| \\ &\text{LPH if} &&\alpha_g + \delta_g < -|\alpha_g - \delta_g| \\ \end{align*}\]
Sample \(\boldsymbol{\theta}\) from \(p(\boldsymbol{\theta} \ | \ \boldsymbol{y})\) by iterating over the Gibbs steps.
\[ \begin{align*} \\ p(\gamma_g^2 | \cdots ) &= \text{Inverse-Gamma} \left ( \frac{N + \nu_\gamma}{2}, \ \frac{1}{2} \left (\nu_\gamma + \sum_{n =1}^N \frac{\varepsilon_{n, g}^2}{\rho_n^2} \right ) \right ) \\ \\ \end{align*} \]
\[ \begin{align*} p(\varepsilon_{n, g} \ | \ \cdots) &\propto \exp \left ( \varepsilon_{n, g}y_{n, g} - \frac{\varepsilon_{n, g}^2}{2 \rho_n^2 \gamma_g^2} - \exp(\varepsilon_{n, g}) \exp(\eta(n, g)) \right) \\ \\ \end{align*} \]
\[ \begin{align*} A &= y_{n, g} \\ B &= \frac{1}{2 \rho_n^2 \gamma_g^2} \\ C &= 0 \\ D &= \exp(\eta(n, g)) \\ E &= 0 \\ \\ \\ \end{align*} \]
\(N \times G = 12 \times 39656\) threads, one for each \(\varepsilon_{n, g}\).
\[ \begin{align*} p(\theta_\phi \mid \cdots) &= \text{Normal} \left ( \frac{ c_\phi^2 \sum_{g = 1}^G \phi_g}{G c_\phi^2 + \sigma_\phi^2}, \ \frac{c_\phi^2 \sigma_\phi^2}{Gc_\phi^2 + \sigma_\phi^2} \right ) \\ p(\theta_\alpha \mid \cdots) &= \text{Normal} \left ( \frac{c_\alpha^2 \sum_{g = 1}^G \alpha_g}{G c_\alpha^2 + \sigma_\alpha^2}, \ \frac{c_\alpha^2 \sigma_\alpha^2}{G c_\alpha^2 + \sigma_\alpha^2} \right ) \\ p(\theta_\delta \mid \cdots) &= \text{Normal} \left (\frac{c_\delta^2 \sum_{g = 1}^G \delta_g}{G c_\delta^2 + \sigma_\delta^2}, \ \frac{c_\delta^2 \sigma_\delta^2}{G c_\delta^2 + \sigma_\delta^2} \right ) \\ \end{align*} \]
\[ \begin{align*} p(\tau_\rho^2 \mid \cdots) &= \text{Gamma} \left (\text{shape} = a_\rho + \frac{N\nu_\rho}{2}, \ \text{rate} = b_\rho + \frac{\nu_\rho}{2} \sum_{n = 1}^N \frac{1}{\rho_n^2} \right ) \\ p(\rho_n^2 | \cdots ) &= \text{Inverse-Gamma} \left ( \frac{G + \nu_\rho}{2}, \ \frac{1}{2} \left (\nu_\rho \tau_\rho^2 + \sum_{g =1}^G \frac{\varepsilon_{n, g}^2}{\gamma_g^2} \right ) \right ) \\ p(\sigma_\phi^2 \ | \ \cdots) &= \text{Inverse-Gamma} \left ( \frac{G - 1}{2}, \ \frac{1}{2} \sum_{g = 1}^G (\phi_g - \theta_\phi)^2 \right ) \text{I} (\sigma_\phi^2 < s_\phi^2) \\ p(\sigma_\alpha^2 \ | \ \cdots) &= \text{Inverse-Gamma} \left ( \frac{G - 1}{2}, \ \frac{1}{2} \sum_{g = 1}^G (\alpha_g - \theta_\alpha)^2 \right ) \text{I} (\sigma_\alpha^2 < s_\alpha^2) \\ p(\sigma_\delta^2 \ | \ \cdots) &= \text{Inverse-Gamma} \left (\frac{G - 1}{2}, \ \frac{1}{2} \sum_{g = 1}^G (\delta_g - \theta_\delta)^2 \right ) \text{I} (\sigma_\delta^2 < s_\delta^2) \end{align*} \]
\[ \begin{align*} p(\nu_\gamma \mid \cdots) \propto \exp &\left ( - \log \Gamma \left ( \frac{\nu_\gamma}{2} \right) + \frac{\nu_\gamma}{2} \log \left ( \frac{\nu_\gamma}{2} \right ) \right . \\ & \left . \ \ - \nu_\gamma \frac{1}{G} \sum_{g = 1}^G \left [ \log \gamma_g + \frac{1}{2} \frac{1}{\gamma_g^2} \right ] \right )^G \times I(0 < \nu_\gamma < d_\gamma) \\ \\ \end{align*} \]
\[ \begin{align*} &\frac{d^2}{d \nu_\gamma^2} \log p(\nu_\gamma \mid \cdots) = - \frac{G}{4} \psi^{(1)} \left ( \frac{\nu_\gamma}{2} \right) + \frac{G}{2 \nu_\gamma} \qquad (0 < \nu_\gamma < d_\gamma) \\ \\ \end{align*} \]
\[ \begin{align*} p(\nu_\rho \mid \cdots) \propto \exp & \left ( - \log \Gamma \left ( \frac{\nu_\rho}{2} \right) + \frac{\nu_\rho}{2} \log \left ( \frac{\nu_\rho \tau_\rho^2}{2} \right ) \right . \\ & \left . \ \ - \nu_\rho \frac{1}{N} \sum_{n = 1}^N \left [ \log \rho_n + \frac{\tau_\rho^2}{2} \frac{1}{\rho_n^2} \right ] \right )^N \times I(0 < \nu_\rho < d_\rho) \\ \\ \end{align*} \]
Red points: 7 genes with Gelman factors slightly above 1.1 for \(\alpha_g\) or \(\delta_g\).
\[\begin{align*} N &= 12 \qquad \qquad &&\nu_\rho = 5 \\ G &= 35,000 \qquad \qquad &&\nu_\gamma = 5 \\ & &&\tau_\rho = 0.1 \\ \\ \theta_\phi &= 3 \qquad \qquad && \sigma_\phi = 1 \\ \theta_\alpha &= 0 \qquad \qquad && \sigma_\alpha = 0.5 \\ \theta_\delta &= 0 \qquad \qquad && \sigma_\delta = 0.5 \end{align*}\]
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